Dynamic control of DNA condensation

Artificial biomolecular condensates are emerging as a versatile approach to organize molecular targets and reactions without the need for lipid membranes. Here we ask whether the temporal response of artificial condensates can be controlled via designed chemical reactions. We address this general question by considering a model problem in which a phase separating component participates in reactions that dynamically activate or deactivate its ability to self-attract. Through a theoretical model we illustrate the transient and equilibrium effects of reactions, linking condensate response and reaction parameters. We experimentally realize our model problem using star-shaped DNA motifs known as nanostars to generate condensates, and we take advantage of strand invasion and displacement reactions to kinetically control the capacity of nanostars to interact. We demonstrate reversible dissolution and growth of DNA condensates in the presence of specific DNA inputs, and we characterize the role of toehold domains, nanostar size, and nanostar valency. Our results will support the development of artificial biomolecular condensates that can adapt to environmental changes with prescribed temporal dynamics.


Supplementary Tables
The sequences of the oligonucleotide strands were designed using NUPACK.We include two spacer bases (TT) at the center of the junction of each nanostar.All sequences in the tables below are listed 5' to 3'.Orthogonal nanostars (Fig. 6) To generate the phase diagram in Figure 3 of the manuscript we used the same parameters as above, but with a non-zero, variable coupling constant .

DNA nanostar variants
In the main paper we consider the reaction term R(x) arising from the introduction of the inhibition and activation reactions.For the inhibition reaction: We don't treat the finite valency found in DNA nanostars, instead the invading molecule completely inactivates the phase separating molecule from participating in phase separation.One could imagine suitable modification of the theory to include the effects of valency by using a more general Flory-Huggins theory with multiple phase separating elements, however in the present model we are most interested in expounding upon the most generic behavior.

Computational Integration
Our mean field equations are solved in Fourier space with periodic boundary conditions using a semi-implicit scheme with in-house C++ code apart from the Fast Fourier Transforms, which are performed using the FFTW library.Analysis of the areas was performed in Wolfram Mathematica.

Effective parameters underpinning Chemical Dissolution
The models we have described are non-equilibrium, therefore in order to analyze the static properties of these systems, we shall proceed by analyzing whether the homogeneous state is stable to small perturbations.The homogeneous state arises from considering the fixed points in the chemical dynamics.Around this state we apply a wavelength dependent perturbation: for a small perturbation around the homogeneous value .By studying how the growth rate is different for different wave vectors we can answer (to first order) whether the homogeneous state is stable.By looking at the system for different parameters, therefore, we can generate a "phase diagram" of the system in the presence of chemical reactions.
Furthermore, we can study the dynamical properties of the system by numerically solving the equations outright, however, prior to doing this it is worth reasoning over the possible dynamics we might expect to arise from chemical disruption There are two different dynamical factors to consider for droplets in this case, growth and decay.It is easy to see that the nature of the growth process is not affected by the attacking chemical, i.e., we should expect the same kinds of coarsening dynamics of droplets to occur whether there is chemical inhibitor or not, as if we introduce chemical inhibitor which is not sufficient to homogenize the system, it will merely inactivate a certain proportion of the monomers, leaving the rest intact.One could then consider the subsystem containing only the active monomers, which would then be identical to the coarsening dynamics of the same system without inhibitor, under the caveat that there will be a small additional effect due to steric repulsion.On the other hand, the process by which droplets decay away has the potential for novel phenomena.In contrast to the standard process of evaporation, which occurs from the surface, the case with an invasive species could have qualitatively different behavior depending on how much the invading species can penetrate into the droplet before it inactivates it.This is something which is qualitatively different dynamically to the ordinary process of raising the temperature.Doing dimensional analysis with this in mind, we wish to consider two different time scales.One timescale is the diffusion time for an inhibitor inside the droplet: where is the effective droplet size and is the diffusion constant of the inhibitor inside the droplet.This is the only way we can obtain a timescale from a length and a diffusion constant which has dimensions This parameter roughly characterizes how long it would take for the inhibitor to diffuse a distance in the droplet.To supplement this we need to consider how quickly the invading reactions occur.We characterize that with the following timescale: where is the rate of reaction of conversion from a phase separating species and inactivating species into an inactivated complex, which would have the following equation in mass action kinetics: The dimensions of the rate are given by .The concentration used in the time scale is the concentration of the phase separating material in the droplet We postulate that the effective qualitative differences in evaporation scenarios arise from the dimensionless parameter: From which we can identify different regimes as a function of this parameter.For instance if then the diffusion time is much shorter than the reaction time, and we would expect that the inactivator principally acts at the surface of the droplet.By contrast, if The inhibitor can diffuse freely through the droplet before it has even had time to react and we might therefore expect that the subsequent dynamics depend more on the volume of the droplet.In full this parameter is given by: In a simulation is a result, not something we can tune.However, all the other parameters we control.If equilibria depends on then we can tune dynamical response by keeping this factor fixed while we change .In actuality, our system has a natural length scale , given by the surface tension, leading to: Non-dimensionalizing the model equations directly leads to the appearance of parameter (not shown).Depending on the regime in which the system is in, the timescales and tell us how quickly they should proceed.So while rescaling both by a factor should leave both unchanged, the process itself should be sped up by a factor .
.2 Nanostars with different arm length Supplementary Figure 2: Effect of toehold on NS featuring long arms.A: 24 bp arm and B: 32 bp arm designs with a toehold for invasion with different lengths of toehold.Condensates do not form when introducing 5 and 7 base long toeholds.Microscopy images representing the condensates after 30 mins of incubation at room temperature (27° C) after anneal process.Scale bars as represented.2.3 Six arm nanostars with adjacent and staggered invasion points after 24 hoursSupplementary Figure3: DNA condensates formation using 6 arm nanostars including 3 toeholds for invasion.A: Adjacent toehold design and B: Staggered toehold design.Each panel includes representative microscopy images of the sample after 360 mins and 24 h incubation after invader addition at room temperature (27° C). [NS] = 5 μM, [I] = 5 μM (or 1X NS).Scale bars are 30 μm.2.7 Condensate regrowth in 4 arm NS with addition of more than one AntiinvaderSupplementary Figure7: Normalized average area following addition of either one or two invaders for the 4 arm NS design at 0.5X and adding respective anti-invader strands.The antiinvasion process for both cases do not have much of a difference up to 60 min.Data corresponds to a single experimental replicate.Error bars were obtained by bootstrapping.